Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 11 (2015), 009, 10 pages
Lagrangian Reduction on Homogeneous Spaces
with Advected Parameters
Cornelia VIZMAN
Department of Mathematics, West University of Timişoara, Romania
E-mail: vizman@math.uvt.ro
Received August 14, 2014, in final form January 22, 2015; Published online January 29, 2015
http://dx.doi.org/10.3842/SIGMA.2015.009
Abstract. We study the Euler–Lagrange equations for a parameter dependent G-invariant
Lagrangian on a homogeneous G-space. We consider the pullback of the parameter dependent Lagrangian to the Lie group G, emphasizing the special invariance properties of the
associated Euler–Poincaré equations with advected parameters.
Key words: Lagrangian; homogeneous space; Euler–Poincaré equation
2010 Mathematics Subject Classification: 53D17; 53D20; 37K65; 58D05; 58D10
1
Introduction
The Euler–Poincaré (EP) equations arise via reduction of the variational principle for a right
G-invariant Lagrangian L : T G → R. With a restricted class of variations, the extremals
of the integral of the reduced Lagrangian ` : g → R correspond to extremals of the original
variational problem for L [9]. The EP equations are written for the right logarithmic derivative
ξ = ġg −1 = δ r g of the curve g in G as
δ`
d δ`
+ ad∗ξ
= 0.
dt δξ
δξ
(1.1)
Here δ`/δξ denotes the functional derivative of `, which depends on the choice of a space g∗ in
δ`
d
duality with g, that is ( δξ
, η) = dt
`(ξ + tη) for all η ∈ g.
t=0
The case of a parameter dependent G-invariant Lagrangian L : T G×V ∗ → R is studied in [2].
The parameter space V ∗ is a linear representation space of the Lie group G and the associated EP
equations include an advection equation for the parameter. These EP equations with advected
parameters are applied to continuum theories in [6]. To integrate complex fluids in this setting,
the case of an affine G-action on the parameter space V ∗ is treated in [4]. The more general
case when the parameter space is a smooth manifold M acted on by G is considered in [5],
applied there to nematic particles. The reduced equations, called EP equations for symmetry
breaking, written for the reduced Lagrangian ` : g × M → R, involve the cotangent momentum
map J : T ∗ M → g∗ :
d δ`
δ`
∗ δ`
+ adξ
=J
.
dt δξ
δξ
δm
In this paper we generalize the Lagrangian reduction with avdected parameters from the Lie
group setting to the homogeneous space setting. We use the approach from [10] that features
the special invariance properties of the reduced equations written for the pullback Lagrangian
to the Lie group G. Starting with a right G-invariant Lagrangian L̄ : T (G/H) → R, the reduced
Lagrangian ` : g → R coming from its pullback to T G must be invariant under the adjoint action
of the Lie subgroup H and under the addition action of its Lie algebra h. As a consequence,
2
C. Vizman
the EP equations (1.1) are now invariant under the following action of the group C ∞ (I, H)
on C ∞ (I, g):
h · ξ = Adh ξ + δ r h.
The geodesic equations for invariant Riemannian metrics on Lie groups (Euler equations)
correspond to reduced Lagrangians ` that are quadratic; a famous example is the ideal fluid
flow as geodesic equations on the group of volume preserving diffeomorphisms [1] (more geodesic
equations on diffeomorphism groups can be found for instance in [11]). The extension of Euler
equations from Lie groups to homogeneous spaces is done in [8].
The plan of the paper is the following. In Section 2 we review a kind of logarithmic derivative for homogeneous spaces. Then we consider parameter dependent G-invariant Lagrangians
on G/H. We treat the case of a linear action on the parameter space in Section 3, where
a multidimensional Hunter–Saxton equation with advected parameter is obtained. We devote
Section 4 to the EP equations for symmetry breaking, obtained for general actions on arbitrary
parameter spaces. We obtain the affine EP equations as a special case. The examples are
mainly on infinite dimensional homogeneous spaces, such as Diff(S 1 )/S 1 , Diff(M )/ Diff vol (M ),
Diff(M )/ Diff iso (M ), and C ∞ (M, G)/G.
2
Logarithmic derivative and EP equations
The Euler–Lagrange (EL) equations associated to right invariant Lagrangians on Lie groups
lead to the Euler–Poincaré (EP) equations involving reduced Lagrangians and written for right
logarithmic derivatives of curves in the Lie group. In this section we recall how this works in
homogeneous spaces of right cosets, following [10].
Given a smooth curve ḡ : I → G/H, we compare the right logarithmic derivatives of two
smooth lifts g1 , g2 : I → G of ḡ. Because there exists a smooth curve h : I → H such that
g2 = hg1 , the logarithmic derivative δ r g2 = Adh δ r g1 + δ r h is obtained from δ r g1 via the left
action of the group C ∞ (I, H) on C ∞ (I, g):
h · ξ = Adh ξ + δ r h.
(2.1)
Hence the right logarithmic derivative for homogeneous spaces is multivalued:
δ̄ r : C ∞ (I, G/H) → C ∞ (I, g)/C ∞ (I, H),
δ̄ r ḡ = C ∞ (I, H) · δ r g,
(2.2)
where g is any lift of ḡ.
The tangent bundle T G of a Lie group G carries a natural group multiplication. In the right
trivialization T G ∼
= g × G the multiplication becomes (η, h)(ξ, g) = (Adh ξ + η, hg). Given a Lie
subgroup H of G, its tangent bundle T H is a subgroup of T G. Let π : G → G/H denote the
canonical projection. Then the surjective submersion T π : T G → T (G/H) is constant on right
T H-cosets of T G and descends to a canonical diffeomorphism between T G/T H and T (G/H).
The following are equivalent data: right G-invariant Lagrangian L̄ on T (G/H), left T Hinvariant and right G-invariant Lagrangian L on T G, as well as reduced Lagrangian ` on g that
is both h-invariant and Ad(H)-invariant. The relation between the Lagrangians is L = L̄ ◦ T π
and we call L the pullback of L̄.
Proposition 2.1 ([10]). If the reduced Lagrangian ` : g → R is h–invariant and Ad(H)invariant, i.e. `(Adh ξ + η) = `(ξ) for all h ∈ H and η ∈ h, then the EP equations
d δ`
δ`
+ ad∗ξ
=0
dt δξ
δξ
are C ∞ (I, H)-invariant for the action (2.1).
(2.3)
Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
3
In other words, equation (2.3) can be seen as an equation for C ∞ (I, H)-orbits in C ∞ (I, g),
i.e. an equation for right logarithmic derivatives (2.2) of curves in the homogeneous space, so
Proposition 2.1 can be reformulated as:
Proposition 2.2 ([10]). A solution of the EL equation for a right G-invariant Lagrangian
L̄ : T (G/H) → R is a curve in G/H such that the logarithmic derivative of one of its lifts to G
satisfies the EP equation (2.3), with ` the reduced Lagrangian of the pullback L : T G → R.
This proposition admits a generalization to Lagrangians that are not necessarily right Ginvariant. First we note the following property of a left T H-invariant Lagrangian L : T G → R:
if the curve g in G is a solution of the corresponding EL equation, then the curve hg is also
a solution of the EL equation, for any smooth curve h in H. Indeed, each variation gε of g with
fixed endpoints corresponds to a variation (ε, t) 7→ h(t)gε (t) of hg with fixed endpoints.
Proposition 2.3. Let L : T G → R be the pullback of the Lagrangian L̄ : T (G/H) → R (i.e. L
is left T H-invariant). Then the following assertions hold:
(i) If the curve g in G is a solution of the EL equation for L, then it descends to the solution
ḡ = π ◦ g of the EL equation for L̄.
(ii) If the curve ḡ in G/H is a solution of the EL equation for L̄, then any lift g of ḡ is
a solution of the EL equation for L.
Proof . Let g be a solution of the EL equation for L and ḡ = π ◦ g. An arbitrary variation ḡε
of ḡ in G/H with fixed endpoints can be lifted to a variation gε of g in G, but it doesn’t
necessarily have fixed endpoints. It only satisfies gε (0) ∈ Hg(0) and gε (1) ∈ Hg(1). We can
achieve gε (0) = g(0) by multiplying gε (t) with g(0)gε (0)−1 from the left. Moreover, we achieve
gε (1) = g(1) by multiplying the new variation gε (t) with g(1)gεt (1)−1 from the left. Now, using
also the identity L = L̄ ◦ T π, we get
Z
Z
d
d
L̄(ḡε (t), g¯˙ε (t))dt =
L(gε (t), g˙ε (t))dt = 0,
dε 0
dε 0
so that ḡ is a solution of the EL equation for L̄. This proves the first assertion.
The second assertion is straightforward, since a variation of g in G with fixed endpoints
always descends to a variation in G/H with fixed endpoints.
A special case is the geodesic equation for a right G–invariant Riemannian metric on G/H,
i.e. Euler equation on homogeneous spaces [8]. The next examples are both of this type.
Example 2.4 ([7]). Let (M, µ) be a volume manifold. The homogeneous space of right cosets
Diff(M )/ Diff vol (M ) is the space of normalized volume forms. The right invariant metric on
Diff(M )/ Diff vol (M ) induced by the degenerate Ḣ 1 inner product on X(M )
Z
hu, vi =
div u div v µ
M
is isometric to the standard L2 metric on an open subset of the sphere of radius 2
the Hilbert space L2 (M ). The isometry is
p
ϕ̄ ∈ Diff(M )/ Diff vol (M ) 7→ 2 Jac(ϕ) ∈ L2 (M ),
p
vol(M ) in
where the Jacobian of ϕ ∈ Diff(M ) is computed w.r.t. µ, i.e. ϕ∗ µ = Jac(ϕ)µ.
The geodesic equation is the multidimensional Hunter–Saxton equation
∂t d(div u) + dLu (div u) + (div u)d(div u) = 0.
(2.4)
4
C. Vizman
R
The reduced Lagrangian `(u) = 12 M (div u)2 µ on X(M ) has the required Diff vol (M )- and
Xvol (M )-invariance properties. For M = S 1 one gets the Hunter–Saxton equation as geodesic
equation on Diff(S 1 )/S 1 .
The left action (2.1) involved in the definition of the right logarithmic derivative δ̄ r on
the homogeneous space Diff(M )/ Diff vol (M ) is the action of the group C ∞ (I, Diff vol (M )) on
C ∞ (I,X(M )) given by
∗
(ψ · u)(t) = ψ(t)−1 u(t) + δ r ψ(t) ∈ X(M )
(2.5)
since Adψ u = (ψ −1 )∗ u. Here the right logarithmic derivative δ r ψ = ∂t ψ ◦ ψ −1 is the time
dependent (divergence free) vector field induced by the (volume preserving) isotopy ψ(t). By
Proposition 2.1 the Hunter–Saxton equation (2.4) is invariant under the action (2.5). This can
be checked also by a direct computation.
Example 2.5 ([7]). Let (M, g) be a Riemannian manifold and Diff iso (M ) its group of isometries.
The homogeneous space of right cosets Diff(M )/ Diff iso (M ) admits a right invariant metric
induced by the degenerate inner product on X(M )
Z
Z
hu, vi =
(Lu g, Lv g) µ =
2 du[ , dv [ + 4 δu[ , δv [ − 4 Ric(u, v) µ.
M
M
R
The reduced Lagrangian `(u) = 21 M |Lu g|2 µ on X(M ) has the required Diff vol (M )- and
Xvol (M )-invariance properties. The associated EP equation
4dδu[t + 2δdu[t − 4 Ric(ut ) + (div u) 4dδu[ + 2δdu[ − 4 Ric(u)
+ Lu 4dδu[ + 2δdu[ − 4 Ric(u) = 0
is invariant under the action (2.5) of C ∞ (I, Diff iso (M )) on C ∞ (I, X(M )).
3
EP equations with advected parameters
Now we look at parameter dependent Lagrangians. First we treat the Lie group case, following [2], then we pass to homogeneous spaces.
3.1
The case of Lie groups
We consider a linear right action ρ of the Lie group G on the vector space V and its dual left
d
action ρ∗ on V ∗ . The corresponding Lie algebra actions on V and V ∗ are dt
ρ
(v) = vξ
0 exp(tξ)
d
∗
r
∗
and dt 0 ρexp(tξ) (a) = ξa. If ξ(t) = δ g(t), then a(t) = ρg(t) (a0 ) is the unique solution of the
differential equation with time-dependent coefficients ȧ = ξa, a(0) = a0 . The diamond operation
: V × V ∗ → g∗ is given by
hv a, ξi := hξa, vi,
for all
ξ ∈ g.
(3.1)
A right G-invariant Lagrangian L : T G×V ∗ → R (including the linear action on the parameter
space in the second argument) has a reduced Lagrangian ` : g × V ∗ → R so that
` vg g −1 , ρ∗g (a) = L(vg , a),
vg ∈ Tg G.
For fixed a0 ∈ V ∗ the Lagrangian La0 : T G → R is right invariant only under the isotropy
subgroup Ga0 of a0 ∈ V ∗ .
Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
5
Theorem 3.1 ([2]). The EL equations for La0 on G given by Hamilton’s variational principle
Z t2
La0 (g(t), ġ(t))dt = 0
δ
t1
can be expressed as EP equations on g × V ∗ with advected parameter:
d δ`
δ`
δ`
+ ad∗ξ
=
a,
dt δξ
δξ
δa
ȧ = ξa
(3.2)
for the reduced Lagrangian `.
The main examples are the heavy top and the ideal compressible fluid. For the heavy top
G = SO(3) and the parameter Γ ∈ V ∗ = R3 is the unit vector in the gravity direction in
body representation. For the ideal compressible fluid G = Diff(M ), with M a Riemannian
manifold, and the parameter ρ ∈ V ∗ = C ∞ (M )∗ is the fluid density in spatial representation.
The reduced Lagrangians
are `(Ω, Γ) = 12 I Ω · Ω − Γ · λ for Ω ∈ so(3) = R3 in the first example,
R
and `(u, ρ) = 21 M |u|2 ρ for u ∈ X(M ) in the second one.
3.2
The case of homogeneous spaces
Let L : T G × V ∗ → R be the pull-back of a G-invariant Lagrangian L̄ : T (G/H) × V ∗ → R,
hence L is left T H-invariant and right G-invariant. If ` : g × V ∗ → R is the reduced Lagrangian,
then
vh ∈ Th H.
`(ξ, a) = L(ξ, a) = L(vh ξ, a) = ` Adh ξ + vh h−1 , ρ∗h (a) ,
This proves the next proposition.
Proposition 3.2. The reduced Lagrangian ` : g × V ∗ → R associated to the pullback of a parameter dependent right G-invariant Lagrangian on G/H is H- and h-invariant:
`(Adh ξ + η, ρ∗h (a)) = `(ξ, a),
h ∈ H,
η ∈ h.
(3.3)
Lemma 3.3. The functional derivatives of the reduced Lagrangian ` : g × V ∗ → R that has the
invariance property (3.3) are equivariant:
δ`
δ`
(Adh ξ + η, ρ∗h (a)) = Ad∗h−1 (ξ, a)
δξ
δξ
and
δ`
δ`
(Adh ξ + η, ρ∗h (a)) = ρh−1 (ξ, a).
δa
δa
Proof . We compute for ζ ∈ g:
d
d
δ`
∗
(Adh ξ + η, ρh (a)), ζ =
`(Adh ξ + η + tζ, ρ∗h (a)) =
`(ξ + t Ad∗h−1 , a)
δξ
dt 0
dt 0
δ`
∗
= Adh−1 (ξ, a), ζ .
δξ
Similarly we get that
δ`
d
d
∗
(Adh ξ + η, ρh (a)), b =
`(Adh ξ + η, ρ∗h (a) + tb) =
`(ξ, a + tρ∗h−1 (b))
δa
dt 0
dt 0
δ`
= ρh−1 (ξ, a), b
δa
for all b ∈ V ∗ .
6
C. Vizman
The path group C ∞ (I, H) acts on C ∞ (I, g × V ∗ ) by
h · (ξ, a) = (Adh ξ + δ r h, ρ∗h (a)).
(3.4)
This action has the property h · (δ r g, ρ∗g a) = (δ r (hg), ρ∗hg a) for any curve g ∈ C ∞ (I, G).
Proposition 3.4. Given a reduced Lagrangian ` : g × V ∗ → R that has the invariance property (3.3), the EP equation with advected parameters (3.2) is C ∞ (I, H)-invariant for the action (3.4).
Proof . We need the G-equivariance of the diamond operation:
Ad∗g (v ρ∗g (a)) = ρg (v) a
that follows from ρ∗g (ξa) = (Adg ξ)(ρ∗g a). Using also the following identities for α ∈ g∗ :
ad∗Adh ξ Ad∗h−1 α = Ad∗h−1 ad∗ξ α,
d
(Ad∗h−1 α) = − ad∗δr h Ad∗h−1 α,
dt
we compute
δ`
d δ`
∗ δ`
+ adξ
−
a (h · (ξ, a))
dt δξ
δξ δa
δ`
δ`
d
δ`
∗
=
Adh−1 (ξ, a) + ad∗Adh ξ+δr h Ad∗h−1 (ξ, a) − ρh−1 (ξ, a) ρ∗h a
dt
δξ
δξ
δa
d
δ`
δ`
δ`
= Ad∗h−1
+ ad∗ξ
−
a (ξ, a).
dt δξ
δξ δa
This ensures the C ∞ (I, H)-invariance of the EP equation with advected parameters.
Example 3.5. Let M be a Riemannian manifold. As in Example 2.4 we focus on the group of
volume preserving diffeomorphisms Diff vol (M ) and the homogeneous space Diff(M )/ Diff vol (M )
of volume forms with constant total volume.
We consider the parameter space C ∞ (M )∗ , identified with C ∞ (M ) via the volume form µ,
hence the left Diff(M )-action and its infinitesimal X(M )-action are
ϕ · ρ = ρ ◦ ϕ−1 Jac(ϕ−1 ),
uρ = −Lu ρ − ρ div u = − div(ρu).
The diamond operation (3.1) becomes
: C ∞ (M ) × C ∞ (M )∗ → X(M )∗ ,
f ρ = ρdf ∈ X(M )∗ ,
where the dual of the space of vector fields is identified via the volume form µ with the space
of differential 1-forms.
The reduced Lagrangian ` : X(M ) × C ∞ (M )∗ → R given by
Z
1
`(u, ρ) =
ρ(div u)2 µ
2 M
comes from a Diff(M )-invariant parameter dependent Lagrangian on Diff(M )/ Diff vol (M ). Indeed, it satisfies the invariance property (3.3): for all ψ ∈ Diff vol (M ) and w ∈ Xvol (M ) we
compute
Z
Z
1
1
−1 ∗ 2
−1 ∗
`(Adψ u + w, ψ · ρ) =
ψ
ρ div(ψ ) u µ =
ρ(div u)2 ψ ∗ µ = `(u, ρ),
2 M
2 M
Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
7
using at step 2 the identity div(ψ ∗ u) = ψ ∗ div u that holds for any volume preserving diffeomorphism ψ.
The EP equation with advected parameters (3.2) becomes
∂t d(ρ div u) + dLu (ρ div u) + d ρ(div u)2 = 0,
∂t ρ + div(ρu) = 0.
By Proposition 3.4 this equation is C ∞ (I, Diff vol (M ))-invariant for the joint action (3.4), namely
∗
∗
ψ · (u, ρ) = ψ −1 u + δ r ψ, ψ −1 (ρ) Jac ψ −1
for curves ψ in Diff vol (M ), u in X(M ), and ρ in C ∞ (M ).
Example 3.6. One can consider as well the group Diff iso (M ) of isometries of M as a subgroup
of Diff(M ), like in Example
2.5. The reduced Lagrangian ` : X(M ) × C ∞ (M )∗ → R would
R
1
be given by `(u, ρ) = 2 M ρ|Lu g|2 µ, coming from a Diff(M )-invariant parameter dependent
Lagrangian on the homogeneous space Diff(M )/ Diff iso (M ).
4
EP equations for symmetry breaking
One can replace the linear action of G on a parameter vector space V ∗ with an arbitrary action
of G on a parameter manifold M . This generalization of the EP equations with advected
parameters, called EP equations for symmetry breaking, are presented in [5]. In this section we
adapt these results to the case of homogeneous spaces.
4.1
The case of Lie groups
Let a Lie group G act on the smooth manifold M from the left, and let ξM ∈ X(M ) denote the
infinitesimal generator of ξ ∈ g. Given a curve g in G starting at the identity, the curve m(t) =
g(t) · m0 is the unique solution of the differential equation with time-dependent coefficients
ṁ = ξM (m),
m(0) = m0 ,
where ξ(t) = δ r g(t).
The cotangent momentum map J : T ∗ M → g∗ , defined by (J(αm ), ξ) = (αm , ξM (m)) for all
∗ M , is G-equivariant for the cotangent and coadjoint actions: J(g · α ) = Ad∗ J(α ).
αm ∈ Tm
m
m
g
Given a right G-invariant Lagrangian L : T G × M → R, i.e.
L vg h, h−1 · m = L(vg , m),
h ∈ G,
its reduced Lagrangian ` : g × M → R satisfies L(vg , m) = `(vg g −1 , g · m). The functional
δ`
δ`
derivative δξ
takes values in g∗ , while δm
is a g-dependent section of T ∗ M .
Theorem 4.1 ([5]). The EL equations for the Lagrangian Lm0 : T G → R are the EP equations
for symmetry breaking
d δ`
δ`
δ`
+ ad∗ξ
=J
,
ṁ = ξM (m)
(4.1)
dt δξ
δξ
δm
for the reduced Lagrangian ` : g × M → R.
Example 4.2 ([5]). For an EP description of nematic particles one considers the canonical
action of G = SO(3) on M = RP 2 . The SO(3)-invariant Lagrangian is
L : T SO(3) × RP 2 → R,
1
λ
2
L(g, ġ, m) = j|ġ|2 − m, g −1 k ,
2
2
where j and λ are constants, and k the external force field, with reduced Lagrangian ` : so(3) ×
RP 2 → R given by `(ξ, m) = 12 j|ξ|2 − λ2 hm, ki2 .
8
4.2
C. Vizman
The case of homogeneous spaces
Let L : T G×M → R be the pull-back of the right G-invariant Lagrangian L̄ : T (G/H)×M → R,
hence L is left T H-invariant (in the first argument) and right G-invariant (in both arguments
simultaneously). The associated reduced Lagrangian ` : g × M → R is both H- and h-invariant:
`(Adh ξ + η, h · m) = `(ξ, m),
h ∈ H,
η ∈ h.
(4.2)
Proposition 4.3. Given a reduced Lagrangian ` : g × M → R that has the invariance property (4.2), the EP equation for symmetry breaking (4.1) is invariant under the C ∞ (I, H)-action
on C ∞ (I, g × T ∗ M ):
h · (ξ, m) = (Adh ξ + δ r h, h · m).
Proof . The equivariance property of the functional derivative
δ`
also the following equivariance property of δm
:
δ`
δξ
from Lemma 3.3 holds, but
δ`
δ`
(h · (ξ, m)) = h−1 ·
(ξ, m).
δm
δm
Indeed, for any curve c in M with c(0) = h · m and c0 (0) = w, we get:
d
d
δ`
(h · (ξ, m)), w =
`(Adh ξ + δ r h, c(t)) =
`(ξ, h−1 · c(t))
δm
dt 0
dt 0
δ`
δ`
−1
−1
=
(ξ, m), h · w = h ·
(ξ, m), w .
δm
δm
Using also the equivariance of the cotangent momentum map, we compute
δ`
d δ`
δ`
δ`
d
δ`
+ ad∗ξ
−J
(h · (ξ, m)) =
Ad∗h−1
+ ad∗Adh ξ+δr h Ad∗h−1
dt δξ
δξ
δm
dt
δξ
δξ
δ`
d δ`
δ`
−1
∗
∗ δ`
−J h ·
= Adh−1
+ adξ
−J
.
δm
dt δξ
δξ
δm
This shows the required invariance of the equation (4.1).
Example 4.4. This is a variation of Example 4.2 for the subgroup H = S 1 of G = SO(3)
consisting of all rotations with axis k. Instead of the reduced Lagrangian `(ξ, m) = 21 j|ξ|2 −
λ
1
λ
2
2
2
2 (m · k) from Example 4.2 one can take `(ξ, m) = 2 j|pk⊥ (ξ)| − 2 hm, ki , where pk⊥ denotes
⊥
3
the orthogonal Euclidean projection onto the vectorial plane k ⊂ R . It has the required
invariance properties because pk⊥ (η) = 0 for all η ∈ h (because η is proportional to k) and
pk⊥ (Adh ξ) = Adh (pk⊥ (ξ)) for all h ∈ H. Indeed, for every rotation h with axis k and every
η ∈ h,
1
λ
1
λ
`(Adh ξ + η, h · m) = j|pk⊥ (Adh ξ)|2 − hh · m, ki2 = j|pk⊥ (ξ)|2 − hm, ki2 = `(ξ, m).
2
2
2
2
4.3
Af f ine EP equations
Now we consider the special case of an affine left G-action on a parameter space V ∗ :
θg (a) = ρ∗g (a) + c(g),
(4.3)
where c : G → V ∗ is a group 1-cocycle for the action ρ∗ , i.e.
c(gh) = c(g) + ρ∗g c(h).
Let dc : g → V ∗ be the associated Lie algebra 1-cocycle. If ξ(t) = δ r g(t), then a(t) = θg(t) (a0 ) is
the unique solution of the differential equation with time-dependent coefficients ȧ = ξa + dc(ξ),
a(0) = a0 .
Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
9
Remark 4.5. Let dc> : V → g∗ be defined by hdc> (v), ξi = hdc(ξ), vi. Then the cotangent
momentum map for the affine action (4.3) of G on V ∗ can be written as
J : T ∗ V ∗ = V ∗ × V → g∗ ,
J(a, v) = v a + dc> (v),
(4.4)
because for all ξ ∈ g,
(J(a, v), ξ) = (v, ξV ∗ (a)) = (v, ξa + dc(ξ)) = v a + dc> (v), ξ .
The following result for a right G-invariant Lagrangian L : T G × V ∗ → R with reduced
Lagrangian ` : g × V ∗ → R is a special case of Theorem 4.1 and a generalization of Theorem 3.1.
Theorem 4.6 ([4]). The EL equations for La0 : T G → R can be expressed as affine EP equations
for the reduced Lagrangian `:
d δ`
δ`
δ`
∗ δ`
>
,
ȧ = ξa + dc(ξ).
(4.5)
+ adξ
=
a + (dc)
dt δξ
δξ
δa
δa
Spin systems. In [4] is shown that the affine EP equations for the action of the gauge group
C ∞ (M, G) on the space V ∗ = Ω1 (M, g) of principal connections on the trivial bundle M × G
θg (γ) = Adg γ − dgg −1
(4.6)
can be used in the description of spin systems. The 1-cocycle is in this case the right logarithmic
derivative
c : C ∞ (M, G) → Ω1 (M, g),
c(g) = −dgg −1 ,
so dc(ξ) = −dξ for all ξ ∈ C ∞ (M, g). The infinitesimal action involves the covariant derivative
dγ ξ = dξ + [γ, ξ], namely ξV ∗ (γ) = −dγ ξ.
We fix a volume form on M , so C ∞ (M, g∗ ) is a dual space to the gauge Lie algebra C ∞ (M, g),
while X(M, g∗ ) is a dual space to the parameter space Ω1 (M, g). The cotangent momentum
map (4.4) becomes J(γ, α) = − ad∗γ α − div α = divγ α, since dc> (α) = div α and that the
diamond map is α γ = − ad∗γ α. We can write now the affine EP equation on C ∞ (M, g) ×
Ω1 (M, g) as
∂ δ`
δ`
δ`
+ ad∗ξ
= − divγ
,
∂t δξ
δξ
δγ
γ̇ + dγ ξ = 0.
(4.7)
For M = R3 and G = SO(3) one gets a macroscopic description of spin glasses [4]. For M a real
interval and G = SE(3), the Euclidean group of rigid motions, one gets an affine EP formulation
of Kirchhoff’s theory of rods (the Cosserat rod) in the case of potential forces [3].
Homogeneous spaces. Let L : T G × V ∗ → R be now the pull-back of a G-invariant
Lagrangian L̄ : T (G/H) × V ∗ → R. Because L is left T H-invariant and right G-invariant, its
reduced Lagrangian ` : g × V ∗ → R is both H- and h-invariant:
`(Adh ξ + η, θh (a)) = `(ξ, a),
h ∈ H,
η ∈ h.
(4.8)
Proposition 4.7. Given a reduced Lagrangian ` : g × V ∗ → R that has the invariance property (4.8), the affine EP equation (4.5) is invariant under the action of the path group C ∞ (I, H)
on C ∞ (I, g × V ∗ ) by
h · (ξ, a) = (Adh ξ + δ r h, θh (a)).
(4.9)
10
C. Vizman
Proof . It is a consequence of Proposition 4.3, but it can be shown also directly, as in the proof
of Proposition 3.4, using the expression of the failure of dc to be G-equivariant: dc(Adg ξ) −
ρ∗g dc(ξ) = c(g) Adg ξ.
Example 4.8 (spin systems). Let G be a Lie group and κ an invariant inner product on its
Lie algebra g. The reduced Lagrangians ` : C ∞ (M, g) × Ω1 (M, g) → R that depend only on the
differential of the function ξ ∈ C ∞ (M, g):
Z
Z
Z
1
1
2
2
`1 (ξ, γ) =
|dξ|2 − |γ|2 µ
|[dξ, γ]| µ,
`2 (ξ, γ) =
|κ(dξ, γ)| µ,
`3 (ξ, γ) =
2
2
all come from a Lagrangian on the homogeneous space C ∞ (M, G)/G because all of them satisfy
the invariance property (4.8). We check it for the middle Lagrangian for all ξ ∈ C ∞ (M, g),
h ∈ G and η ∈ g (so dhh−1 = 0 and dη = 0):
Z
2
(4.6) 1
`2 (Adh ξ + η, θh (γ)) =
κ Adh dξ + dη, Adh γ − dhh−1 µ = |κ(dξ, γ)|2 µ = `2 (ξ, γ).
2
It follows that the corresponding EP equations (4.7) for spin systems are invariant under the
action (4.9), hence (4.7) can be seen as an equation on C ∞ (M, G)/G.
This setting of affine EP reduction is used in [12] for the dynamical description of spacetime strands on homogeneous spaces. Covariant EP equations on homogeneous spaces provide
another frame to describe the dynamics of space-time strands on homogeneous spaces.
Acknowledgements
The author is grateful to the referee for very helpful suggestions. This work was supported by
a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, project
number PN-II-ID-PCE-2011-3-0921.
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